Sampling (DP IB Maths: AI HL)

What are the different types of data?

•  Qualitative data is data that is usually given in words not numbers to describe something
  • For example: the colour of a teacher's car
• Quantitative data is data that is given using numbers which counts or measures something
  • For example: the number of pets that a student has
• Discrete data is quantitative data that needs to be counted
  • Discrete data can only take specific values from a set of (usually finite) values
  • For example: the number of times a coin is flipped until a ‘tails’ is obtained
• Continuous data is quantitative data that needs to be measured
  • Continuous data can take any value within a range of infinite values
  • For example: the height of a student
• Age can be discrete or continuous depending on the context or how it is defined
  • If you mean how many years old a person is then this is discrete
  • If you mean how long a person has been alive then this is continuous

What is the difference between a population and a sample? 

• The population refers to the whole set of things which you are interested in
  • For example: if a vet wanted to know how long a typical French bulldog slept for in a day then the population would be all the French bulldogs in the world
• A sample refers to a subset of the population which is used to collect data from
  • For example: the vet might take a sample of French bulldogs from different cities and record how long they sleep in a day
• A sampling frame is a list of all members of the population
  • For example: a list of employees’ names within a company
• Using a sample instead of a population:
  • Is quicker and cheaper
  • Leads to less data needing to be analysed
  • Might not fully represent the population
  • Might introduce bias

What is a random sample and a biased sample? 

• A random sample is where every member of the population has an equal chance of being included in the sample
• A biased sample is one from which misleading conclusions could be drawn about the population
  • Random sampling is an attempt to minimise bias

What sampling techniques do I need to know? 

Simple random sampling 

• Simple random sampling is where every group of members from the population has an equal probability of being selected for the sample
• To carry this out you would...
  • uniquely number every member of a population
  • randomly select n different numbers using a random number generator or a form of lottery (where numbers are selected randomly)
• Effectiveness:
  • Useful when you have a small population or want a small sample (such as children in a class)
  • It can be time-consuming if the sample or population is large
  • This can not be used if it is not possible to number or list all the members of the population (such as fish in a lake)

Systematic sampling 

• Systematic sampling is where a sample is formed by choosing members of a population at regular intervals using a list
• To carry this out you would...
  • calculate the size of the interval 
  • choose a random starting point between 1 and k
  • select every kth member after the first one
• Effectiveness:
  • Useful when there is a natural order (such as a list of names or a conveyor belt of items)
  • Quick and easy to use
  • This can not be used if it is not possible to number or list all the members of the population (such as penguins in Antarctica)

Stratified sampling 

• Stratified sampling is where the population is divided into disjoint groups and then a random sample is taken from each group 
• The proportion of a group that is sampled is equal to the proportion of the population that belong to that group
• To carry this out you would...
  • Calculate the number of members sampled from each stratum
    • 
  • Take a random sample from each group
• Effectiveness:
  • Useful when there are very different groups of members within a population
  • The sample will be representative of the population structure
  • The members selected from each stratum are chosen randomly
  • This can not be used if the population can not be split into groups or if the groups overlap

Quota sampling 

• Quota sampling is where the population is split into groups (like stratified sampling) and members of the population are selected until each quota is filled
• To carry this out you would...
  • Calculate how many people you need from each group
  • Select members from each group until that quota is filled
    • The members do not have to be selected randomly
• Effectiveness:
  • Useful when collecting data by asking people who walk past you in a public place or when a sampling frame is not available
  • This can introduce bias as some members of the population might choose not to be included in the sample

Convenience sampling

• Convenience sampling is where a sample is formed using available members of the population who fit the criteria
• To carry this out you would...
  • Select members that are easiest to reach
• Effectiveness:
  • Useful when a list of the population is not possible
  • This is unlikely to be representative of the population structure
  • This is likely to produce biased results

What are the main criticisms of sampling techniques? 

• Most sampling techniques can be improved by taking a larger sample
• Sampling can introduce bias - so you want to minimise the bias within a sample
  • To minimise bias the sample should be as close to random as possible
• A sample only gives information about those members
  • Different samples may lead to different conclusions about the population

How can I decide if data is reliable?

• Data from a sample is reliable if similar results would be obtained from a different sample from the same population
• The sample should be representative of the population
• The sample should be big enough
  • Sampling a small proportion of a population is unlikely to be reliable

What can cause data to be unreliable?

• If the sample is biased
  • It is not random
• If errors are made when collecting data
  • Numbers could be recorded incorrectly, duplicated or missed out
• If the person collecting the data favours some members over others
  • They might seek out members who will lead to a desired outcome
  • They might exclude members if they would cause the sample to oppose the desired outcome
• If a significant proportion of data is missing
  • Some data may be unavailable
  • Some members might decide not to be part of the sample
    • This will mean the results are not necessarily representative of the population